User talk:MathMartin/Archive1

Welcome!

I realise this is a bit belated, but I just wanted to welcome you to Wikipedia, and invite you to join Wikipedia: WikiProject Computing and Wikipedia: WikiProject Mathematics if you have not already done so. I also have a strong interest (and a degree) in both computer science and math, and I invite you to drop me any questions you might have, or to mercilessly assault any of the pages listed on my user page. Thanks for editing, and I hope you've come to stay. Deco 00:05, 17 Nov 2004 (UTC)

Discrete

You wrote:

I am pretty sure (albeit have no prove or reference) that all problems in a small event space can be treated combinatoricly or by counting. Why do you think a small event space is called discrete ?

But the Poisson distribution is a discrete distribution. Showing how it arises as a limit of binomial distributions is not just counting. Michael Hardy 23:55, 9 Nov 2003 (UTC)

You also asked:

Why do you think a small event space is called discrete ?

This seems to presuppose that "discrete" means "pertaining to combinatorial enumeration". I am surprised; I didn't know some people thought it meant only that. Discrete topological spaces are called discrete topological spaces for reasons clearly in conflict with such a presupposition, and yet the name "discrete" obviously fits them very well. What about discrete probability distributions that assign positive probability to every rational number? The obviously exist and they are obviously discrete, since all of the probability is accounted for by point-masses, but how would one justify a claim that the reason they're called "discrete" has something to do with combinatorial enumeration? Michael Hardy 03:32, 11 Nov 2003 (UTC)

This seems to presuppose that "discrete" means "pertaining to combinatorial enumeration".

I think this is the meaning of discrete. Discrete math as the science of counting as opposed to continouus math using analysis.

What about discrete probability distributions that assign positive probability to every rational number? The obviously exist and they are obviously discrete, since all of the probability is accounted for by point-masses, but how would one justify a claim that the reason they're called "discrete" has something to do with combinatorial enumeration?

To justify my claim, you would have to show me a problem in a discrete space, and I would point out to you (hopefully) that the method used in solving the problem are combinatorically in nature. So discrete by your definition means 'point-masses' ? I think discrete means countable.

Discrete topological spaces are called discrete topological spaces for reasons clearly in conflict with such a presupposition, and yet the name "discrete" obviously fits them very well.

I can't really comment on this. I have looked it up but it seems to me 'discrete' there has a different meaning.

In any case there is a certain difference between working in a discrete probability space as opposed to working in a continouus one. And the difference in the space should lead to a difference in the mathematics used when working in the space.

How would you characterize the mathematics used in a discrete space as opposed to those used in a continouus one ?

On a side note how do you automatically insert the date like '23:55, 9 Nov 2003 (UTC)' ? User:MathMartin

Signature with date and time is done with ~~~~ (quadruple tilde). --Zundark 20:13, 14 Nov 2003 (UTC)

---

What I was trying to say is this:

• finite probability space -> mainly combinatorics
• countable but not finite -> ???
• uncountable probability-space -> mainly measure theory

countable (finite and non finite) probability spaces are called discrete, uncountable probability-spaces are called continouus.

Do you agree on this categorization ? MathMartin 23:07, 14 Nov 2003 (UTC)

Curves

Hello. I appreciate your message. I'm not interested, however, in contributing to the mathematics articles anymore. I have philosophic disagreements with some editors and have found working with some of them to be unrewarding (I'm probably too "old school" to write wiki articles in my professional field, so I've chosen to write about topics are purely hobbies). But I wish you luck. You're certainly welcome, of course, to do whatever you please with my previous contributions. Best wishes. -- Decumanus | Talk 22:07, 14 May 2004 (UTC)

typo?

Hello. The "typo" you fixed in Cauchy-Schwarz inequality was not a typo; it was correct as it stood and you made in incorrect. Sometimes you need to be careful. Michael Hardy 22:55, 18 May 2004 (UTC)

Ok, next time I try to be more careful.

MathMartin 23:06, 18 May 2004 (UTC)

I thought it was good style in English to capitalize the headlines ?

MathMartin 23:15, 18 May 2004 (UTC)

It's not conventional on Wikipedia. I'll find the page with that convention; stand by .... Michael Hardy 23:54, 18 May 2004 (UTC)

OK, here it is: Wikipedia:Manual_of_Style_(headings). Michael Hardy 23:56, 18 May 2004 (UTC)

Replied on my talk page. Dysprosia 00:08, 19 May 2004 (UTC)

Style

Thank you for the vote of confidence.

In several articles on combinatorics I've written "In [[combinatorics | combinatorial]] [[mathematics]]". Maybe "In the [[mathematics | mathematical]] discipline of [[linear algebra]], a triangular matrix is ..." would be a good way to express what you have in mind. Michael Hardy 20:00, 1 Jul 2004 (UTC)

named for vs. named after

Hi Michael, I have a short question on English language. Whats the difference between the following two phrases ?

1. The theorem is named after Michael Hardy
2. The theorem is named for Michael Hardy

Thanks. MathMartin 15:02, 24 Jul 2004 (UTC)

Not sure, except that the second is a bit alien to me. Michael Hardy 18:36, 24 Jul 2004 (UTC)

To clarify further for you and any other non-native English speakers: Both forms are used. This may be merely regional or personal variation in dialect or idiolect. Either will be understood. (I learned "named after" as a child, and "named for" has always been slightly alien to me.) In some articles, I've written named in honor of. Michael Hardy 18:58, 28 Jul 2004 (UTC)

I agree with Michael: "after" is more common and seems more natural to me, but "for" is perfectly acceptable. Adking80 02:00, 26 Mar 2005 (UTC)

imaginary unit

Hello. I've just written it like this:

$e^{it}=\cos(t)+i\sin(t).$

Michael Hardy 16:32, 6 Aug 2004 (UTC)

Diff. Geom. of curves

Re: your post on my talk page. Hi. In my experience (international: math work in the UK and USA), one uses $\to$ when talking about a mapping between two domains and one uses $\mapsto$ to signify what the mapping does to a particular element, e.g. the square function

$\mathrm {sq} :\mathbb {R} \to \mathbb {R}$

$x\mapsto x^{2}.$

I'm almost certain this is the convention for these things, but I'm a little fuzzy on that at the moment. Best, mat_x 20:54, 26 Aug 2004 (UTC)

I emphatically second the comments above (except that I'm not "fuzzy" about this point). Michael Hardy 22:19, 28 Aug 2004 (UTC)

Proposed TeX style guide

I think I could contribute a number of things to such a style guide, but I'm not prepared to actually start the thing. Michael Hardy 22:19, 28 Aug 2004 (UTC)

Nu as index

I'm afraid I disagree on this one. Of course a more experienced mathematically oriented reader will have no trouble whatsover in picking nu up as a subscript, but:

• it's idiosyncratic - no text I've encountered uses nu in this way. Using a typical subscript such as

i,j,k,s,t, or whatever, is much more lucid. Compare $\sum _{\xi =0}^{\Xi }x_{\xi }$ to $\sum _{i=0}^{n}x_{i}$.

• it's not completely accessible to all readers, it may be jarring to inexperienced readers to have a Greek latter as a subscript

• your statement that the "hierachy of indices" makes things clearer, may make things more confusing to some as well (ie "too many n's")

I don't know how we can proceed on this one. Dysprosia 10:47, 17 Sep 2004 (UTC)

it's idiosyncratic - no text I've encountered uses nu in this way. Using a typical subscript such as i,j,k,s,t, or whatever, is much more lucid. Compare $\sum _{\xi =0}^{\Xi }x_{\xi }$ to $\sum _{i=0}^{n}x_{i}$.

I think the first example is a bit better (depending on the context used) but I agree the second example is more common. Even better than both would be

$\sum _{n=0}^{N}x_{n}$.

The numerical analysis script I am currently reading uses $\nu$ and I guess most numerical math text do as they need more subscripts. Have you ever seen $\zeta$ to denote the zero of a polynomial ? I think this is a similar case as the $\zeta$ is used because zero starts with z and in the same way if I do a sum I would use $\nu$ because the upper limit of the sum is denoted by n.

it's not completely accessible to all readers, it may be jarring to inexperienced readers to have a Greek latter as a subscript

This is true as I already admitted.

your statement that the "hierachy of indices" makes things clearer, may make things more confusing to some as well (ie "too many n's")

I do not understand how they make things more confusing. If you have two sums

$\sum _{n=0}^{N}\sum _{\nu =0}^{n}a_{\nu ,n}$

and

$\sum _{m=0}^{M}\sum _{\mu =0}^{m}a_{\mu ,m}$

the m's and n's provide a hint which subscript belongs to which sum.

As I already said I am not interested in reverting your changes so I see no problem. I even agree to your change on the polynomial page. But have a look at Newton polynomial (even on the off chance you would change the subscripts there). Do you really think in this case your convention would make the page clearer ? MathMartin 11:28, 17 Sep 2004 (UTC)

Replied on my talk page. Dysprosia 13:58, 17 Sep 2004 (UTC)

Chebyshev polynomials

Hi, we have met on a few numerical analysis pages. If you have any spare time it would nice if you could check Chebyshev polynomials and especially Chebyshev polynomials#Polynomial interpolation for errors.MathMartin 20:55, 15 Sep 2004 (UTC)

Thanks for the hint; I am always happy to review pages I know (or should know) something about. Thans also for your efforts on interpolation and related pages. As you perhaps noticed, I made some changes to the article on Chebyshev polynomials (as always, feel free to revert), but everything was correct as far as I could see. I also looked at Chebyshev nodes (which you probably edited only after you wrote the above text) and made some minor changes there, but I want to go back later and make some more changes. In particular, it is not clear to me why you introduce p_n, the polynomial of degree n which approximates a given function best in the maximum norm. -- Jitse Niesen 23:09, 17 Sep 2004 (UTC)

PS: I forgot where you asked this, but do make the change from t to x. -- Jitse Niesen 23:16, 17 Sep 2004 (UTC)

I rewrote Chebyshev nodes but I am not really satisfied. Feel free to improve my notation. More generally I intend to rewrite most of the spline pages which are in a really bad shape and contribute to many article on numerical analysis in the next few weeks. As the material is new to me I will probably make some mistakes and often lack the necessary broader scope. It would be nice if you could keep a watchful eye on me.MathMartin 11:07, 19 Sep 2004 (UTC)

You seem to be writing faster than I can check! This is of course good as many numerical analysis pages are indeed far worse than I'd like. In my opinion, the main problem with Wikipedia is not that we don't have enough articles, but that they are not good enough, but that's another issue. I don't know that much about splines, but I'll do my best.

I made some changes in Chebyshev nodes. Two comments:

1. Where did you get the ||·||₀ notation for maximum norms from? I'd use ||·||_∞ (see e.g. Lp space), but perhaps I'm too theoretically minded. Similarly, for me the space C₀[−1, 1] is the space of continuous functions f with f(−1) = f(1) = 0.
2. I removed the sentence "The Chebyshev nodes are important in approximation theory because when used as nodes in polynomial interpolation the resulting interpolation polynomial provides the best approximation to a continuous function under the maximum norm." This might be taken to imply that given a function f, the polynomial p of a given degree that minimizes ||f−p||₀ is constructed by doing interpolation at the Chebyshev nodes, which is not true.

Hopefully, I'll soon find time to write a bit about the Lebesgue constant, which seems to be the thing you were leading up to. O yes, one last note: I personally think that references should be provided with every article, to help the reader and also out of honesty. You don't need to give a references for every statement (though I wouldn't mind if you did!). Cheers, Jitse Niesen 21:31, 19 Sep 2004 (UTC)

Runge phemonena

Hi, I have a question concerning Runge's phemonena. When using Chebyshev nodes to interpolate a function we can minimize the interpolation error, but the interpolation error still increases when we increase the degree of the polynomial. Is this true ?MathMartin 19:58, 20 Sep 2004 (UTC)

That depends on the function that you are interpolating. There are continuous functions for which the interpolation error still increases with the degree of the polynomial when using Chebyshev nodes. However, for "suitably nice" functions, the interpolation error will go to zero as the degree goes to infinity. I don't quite remember what "suitably nice" is, but I think that differentiable (or maybe continuously differentiable) is enough. On the other hand, if you use equidistant nodes, the interpolation may increase even if the function is analytic. Groetjes, Jitse Niesen 20:33, 20 Sep 2004 (UTC)

Different matrix notation

Do you know of any difference between the matrix notation

${\begin{bmatrix}1&0\\0&5\end{bmatrix}}$

and

${\begin{pmatrix}1&0\\0&5\end{pmatrix}}$

Which one is more common ? MathMartin 17:28, 25 Sep 2004 (UTC)

As far as I know, there is no difference. I have no idea which one is more popular; it's just a personal decision which one to use. -- Jitse Niesen 17:54, 26 Sep 2004 (UTC)

I believe that the square brackets are more common in applied mathematics such as numerical analysis, whereas the round brackets are more common in pure mathematics, e.g. abstract algebra. My textbooks seem to agree. Adking80 02:00, 26 Mar 2005 (UTC)

Function problem

In computational complexity theory, "decision problem" is a technical term. Garey & Johnson, Computers and Intractability, page 13: "A decision problem is one whose solution is either "yes" or "no"." Papadimitriou, Computational Complexity, page 3: "REACHABILITY asks a question that requires either a "yes" or "no" answer. Such problems are called decision problems." Gdr 16:35, 2004 Nov 19 (UTC)

So far I agree on your definition of decision problem. (1) Do you agree that a decision problem is a problem with a 1-bit answer (if you call it yes/no or true/false does not matter)? (2) A function problem is like a decision problem but the answer is more complex. (3) So one could say a function problem is a decision problem with a n-bit answer. (4) Every decision problem can be treated as a function problem with a 1-bit answer. (5) Every function problem can be rephrased into n separate decision problems. MathMartin 18:12, 19 Nov 2004 (UTC)

(1) yes (2) yes (3) no (4) yes (5) yes Gdr 18:57, 2004 Nov 19 (UTC)

Ok, I think I understand. I did not meant to say a function problem is a decision problem, but is like a decision problem (only with a more complex answer). Is this our missunderstanding ? MathMartin 19:05, 19 Nov 2004 (UTC)

Perhaps. Gdr 21:25, 2004 Nov 19 (UTC)

I think I've covered your disagreement on the decision problem page, coincidentally. I see you pretty much already have above as well. A function problem is simply a computation problem. If there is a function problem page, I suggest it should be redirected to the computation problem page and probably conjoined. I was going to suggest that you, Martin, work on the computation problem page since it's only a stub.Nortexoid 02:11, 23 Nov 2004 (UTC)

Article Licensing

Hi, I've started a drive to get users to multi-license all of their contributions that they've made to either (1) all U.S. state, county, and city articles or (2) all articles, using the Creative Commons Attribution-Share Alike (CC-by-sa) v1.0 and v2.0 Licenses or into the public domain if they prefer. The CC-by-sa license is a true free documentation license that is similar to Wikipedia's license, the GFDL, but it allows other projects, such as WikiTravel, to use our articles. Since you are among the top 1000 Wikipedians by edits, I was wondering if you would be willing to multi-license all of your contributions or at minimum those on the geographic articles. Over 90% of people asked have agreed. For More Information:

• Multi-Licensing FAQ - Lots of questions answered
• Multi-Licensing Guide
• Free the Rambot Articles Project

To allow us to track those users who muli-license their contributions, many users copy and paste the "{{DualLicenseWithCC-BySA-Dual}}" template into their user page, but there are other options at Template messages/User namespace. The following examples could also copied and pasted into your user page:

Option 1

I agree to [[Wikipedia:Multi-licensing|multi-license]] all my contributions, with the exception of my user pages, as described below:

{{DualLicenseWithCC-BySA-Dual}}

OR

Option 2

I agree to [[Wikipedia:Multi-licensing|multi-license]] all my contributions to any [[U.S. state]], county, or city article as described below:

{{DualLicenseWithCC-BySA-Dual}}

Or if you wanted to place your work into the public domain, you could replace "{{DualLicenseWithCC-BySA-Dual}}" with "{{MultiLicensePD}}". If you only prefer using the GFDL, I would like to know that too. Please let me know what you think at my talk page. It's important to know either way so no one keeps asking. -- Ram-Man (comment| talk)

Re: Categorization

You are of course right to point out that K-theory shouldn't be in a catergory and sub-catergory. (Even though it was already in both Algebra and one of its sub-catergory's, Algebraic topology). My reason for having it outside Alegbra was becuase it one of those areas that cut across several major branches of mathemetics, being related to algebra, geometry and applied maths through its uses in string theory. Consequently, I feel it should either remain in Mathematics, or be listed in all the numerous sub-catergories to which it could fall into. What do you think? 20:00, 29 Dec 2004 (UTC) (sorry, forgot to add that)